Evgenij Semenov

Haar series and linear operators

1. Preliminaries. 2. Definition and Main Properties of the Haar System. 3. Convergence of Haar Series. 4. Basis Properties of the Haar System. 5. The Unconditionality of the Haar System. 6. The Paley Function. 7. Fourier-Haar Coefficients. 8. The Haar System and Martingales. 9. Reproducibility of the Haar System. 10. Generalized Haar Systems and Monotone Bases. 11. Haar System Rearrangements. 12. Fourier-Haar Multipliers. 13. Pointwise Estimates of Multipliers. 14. Estimates of Multipliers in L1. 15. Subsequence of the Haar System. 16. Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces. 17. Olevskii System. References. Index.

Uncomplemented subspaces of Lorentz spaces

In the paper we construct a system of bounded functions which generates an uncomplemented subspace in the Lorentz space Λ(α) for all α∈(0,1). Lower bounds of the norms of the projector onto such subspaces are obtained.

Fourier-Haar Multipliers

In this section we investigate multipliers with respect to the H.s. A sequence λ n , k , (n, k) ∈ Ω generates the operator

Basis Properties of the Haar System

\Lambda (\sum\limits_{(n,k) \in \Omega } {{a_{n,k}}x_n^k} ) = \sum {{\lambda _{n,k}}{c_{n,k}}x_n^k}

Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces

(1) on the polynomials with respect to the H.s. Such operators are said to be multipliers. Recall that the norm of Λ from L p into L p (L p ) is denoted by ‖Λ‖p,q (‖Λ‖ p ). The main result of Chapter 5 (Corollary 5.8) may be formulated in the following way. If , then (2)

Convergence of Haar Series

We prove that the Zygmund space L(lnL)1/2 is the largest among symmetric spaces X in which any uniformly bounded orthonormal system of functions contains a sequence such that the corresponding space of Fourier coefficients F(X) coincides with ℓ2. Moreover, we obtain a description of spaces of Fourier coefficients corresponding to appropriate subsequences of arbitrary uniformly bounded orthonormal systems in symmetric spaces located between the spaces L(lnL)1/2 and L1.

The Haar system and martingales

Theorem 3.2 shows that the H.s. forms a basis in L p , 1 ≤ p < ∞. This statement may be generalized.

Reproducibility of the Haar system

The purpose of this chapter is to describe monotone bases in r.i. spaces. If any contractive projection P satisfying the condition Pk (0,1) = k (o,1) is a conditional expectation, then such description can be given in terms of generalized Haar systems. We start in section 10.a with the characterization of r.i. spaces with the above mentioned property.